Question

a rectangular lot is 125 yards long and 80 yards wide. give the length and width of another rectangular lot that has the same perimeter but a smaller area. width = yards length = yards

Explanation:

Step1: Calculate the perimeter of the original rectangle

The formula for the perimeter of a rectangle is \(P = 2(l + w)\).
For the original rectangle with \(l = 125\) yards and \(w = 80\) yards, \(P=2(125 + 80)=2\times205 = 410\) yards.

Step2: Express the perimeter formula for the new rectangle

Let the length of the new rectangle be \(x\) and the width be \(y\). Then \(2(x + y)=410\), so \(x + y=205\), and \(y = 205 - x\).

Step3: Calculate the area formula for the new rectangle

The area formula for a rectangle is \(A=xy=x(205 - x)=205x - x^{2}\).
We know that for a quadratic function \(A(x)=-x^{2}+205x\) (where \(a=-1\), \(b = 205\), \(c = 0\)), the vertex of the parabola \(y = ax^{2}+bx + c\) has \(x=-\frac{b}{2a}\). Here \(x=\frac{-205}{2\times(-1)} = 102.5\), and the maximum - area occurs when \(x = 102.5\) (a square - like shape for a rectangle). To get a smaller area, we can choose values of \(x\) and \(y\) that are further from each other.
Let's choose \(x = 150\) (we can choose other values as long as \(x + y=205\)). Then \(y=205 - 150=55\).

Answer:

width = \(55\) yards, length = \(150\) yards